I have a doubt about the completeness proof of propositional logic that appears in Chiswell and Hodges´ book. In page 93, case 2, i) I don´t understand why I can replace χ_1 by χ_1 and χ_2, because I don´t know if the rules for and introduction are followed in derivation D. I also don't know what I accomplish by making such a replacement, because χ_1 and χ_2 are in D and clearly I can put after χ_1 and χ_2. Also the book says the result is a derivation of ⊥ whose undischarged assumptions all lie in Γk , but I think that these undischarged assumptions all lie in in Γk+1, since I put in D χ_1 and also χ_2.
I am not a mathematician, I am sorry if the question is very easy.